Question

which statements are true about a 45° - 45° - 90° triangle? the hypotenuse is √3 times the length of a leg. the hypotenuse is √2 times the length of a leg. the triangle is scalene. both legs are congruent. the triangle is isosceles. the triangle is equilateral.

Explanation:

Step1: Recall the properties of a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle

Let the length of each leg be \(a\). By the Pythagorean theorem \(c^{2}=a^{2}+a^{2}\), where \(c\) is the hypotenuse.

Step2: Simplify the Pythagorean equation

\(c^{2}=2a^{2}\), so \(c = a\sqrt{2}\). This shows the hypotenuse is \(\sqrt{2}\) times the length of a leg.

Step3: Analyze the side - congruence

Since the two non - hypotenuse sides (legs) are equal (\(a=a\)), the triangle is isosceles (a triangle with at least two congruent sides). A scalene triangle has all sides of different lengths, and an equilateral triangle has all sides equal (each angle \(60^{\circ}\) in an equilateral triangle).

Answer:

The hypotenuse is \(\sqrt{2}\) times the length of a leg; Both legs are congruent; The triangle is Isosceles.